If a large amount of air (roughly 100ml injected very very quickly) enters your heart it can be locked down and fail to transport blood which will obviously lead to death.
However, it only takes about 2cc of air to kill you and much less to cause a heart attack if you have a patent foramen ovale and are unlucky. The foramen ovale is an opening between the atria of the heart which allows blood to flow from the right atrium to the left atrium. It usually closes at birth when your lungs start working. However in about 1 out 4 people it stays open.
Venous blood from your systemic circulation enters the right atrium and can (in the case of a PFO) then enter the left atrium from where it can again enter the arterial part of the systemic circulation through the left ventricle and the aorta, instead of entering the pulmonary circulation through the right ventricle as it is supposed to.
Regularly whichever amount of air enters the arterial part of the pulmonary circulation can simply be breathed off. No harm done.
In the case of air being pushed out into the arterial part of the systemic circulation it will end up in small arterioles (and proximally of those), causing the bloodflow to the tissue that the respective arterioles supply to stop and in turn causing said tissue to die.
Now in a best case scenario this will mean some random damage in the body that you might not even notice. In a worst case scenario the air will enter your coronary arteries (which supply the heart itself with blood). A very small amount of air can cause a heart attack there. Again very small heart attacks will go by unnoticed, but 2ml might kill you. The same goes
If you therefore have such a PFO it could happen that every few times you inject a tiny bubble you are causing a tiny heart attack or a tiny stroke or other tiny damages throughout your body... That being said, it's best to get rid of that air in the syringe!!
On a funny side note, check out this women who died due to aspirating air through her vagina while masturbating by inserting a carrot:
http://www.ncbi.nlm.nih.gov/pubmed/11453096